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Matthew Cushman : The Motivic Fundamental Group

The fundamental group of a topological space is usually defined in terms of homotopy classes of based loops. The group structure is given by composition of loops. If X is a complex algebraic variety, one has an underlying topological space, and hence a fundamental group. Hain showed that the nilpotent completion of the group ring of this topological fundamental group carries a mixed Hodge structure. Grothendieck defined a fundamental group for schemes defined over any field. Applying this to a complex algebraic variety, one obtains the profinite completion of the topological fundamental group. This group comes with a natural action of the absolute Galois group of the field of definition. The above indicates that varieties over fields of characteristic zero have two notions of fundamental group, armed with either a Galois action or a mixed Hodge structure. This is similar to the situation with homology and cohomology groups, where one has both an etale and Betti version carrying Galois actions and Hodge structures. An important guiding principle is that both of these versions of homology and cohomology should come from an underlying ``motivic'' theory. This is a homology and cohomology theory for algebraic varieties over a field k taking values the abelian tensor category of mixed motives over k, denoted M_k. There should be functors from M_k to both the category of Galois modules and mixed Hodge structures. When applied to the motivic homology of a variety X, these functors should yield the etale homology or Betti homology of X. In this way, motives glue these two different theories together more strongly than just the comparison isomorphisms. Nori has recently given a definition of the category of mixed motives. In this talk, we will show how this category relates to the fundamental group. In fact, more generally there is a motivic version of paths between two different points x and y of X which is important for applications. We also show that the multiplication and comultiplication maps are motivic, and compare with Hain's theory.

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